
Let \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\)  be the category fibered in groupoids over \(\operatorname{\bf {RPC}}\)  such that:

the objects are triples \((\Gamma /\sigma , D)\) , where \(\Gamma /\sigma \)  is a quasi-stable tropical curve over \(\sigma \in \operatorname{\bf {RPC}}\) with \(n\) -markings and of genus \(g\)  (or simply of type \((g,n)\) ), i.e. a pair consisting of a quasi-stable graph \((\Gamma )\)  of type \((g,n)\)  (called the underlying graph) and a (generalized) metric \(d:=d_{\Gamma }:E((\Gamma ))\rightarrow S_{\sigma }\setminus \lbrace 0\rbrace \) ,
and \(D\)  is an admissible divisor on \((\Gamma )\)  (so that \(((\Gamma ),D)\in \operatorname{\mathcal {QD}iv}_{g,n}\) );

the morphisms are pairs \((f,\pi ):( \Gamma /\sigma , D)\rightarrow (\Gamma ^{\prime }/\sigma ^{\prime },D^{\prime })\)  such that \(f:\sigma \rightarrow \sigma ^{\prime }\)  is a morphism in \(\operatorname{\bf {RPC}}\)  and \(\pi :((\Gamma ^{\prime }),D^{\prime })\rightarrow ((\Gamma ),D)\)  is a morphism in \(\operatorname{\mathcal {QD}iv}_{g,n}\) 
such that \(\pi \)  is compatible with \(f\)  as in [1]}, i.e. such that for any \(e^{\prime }\in E((\Gamma ^{\prime }))\)  we have that

\(\pi \)  contracts \(e^{\prime }\)  if and only if \(f^*(d_{\Gamma ^{\prime }}(e^{\prime }))=0\) ;

if \(\pi (e^{\prime })=e\in E((\Gamma ))\)  then \(f^*(d_{\Gamma ^{\prime }}(e^{\prime }))=d_{\Gamma }(e)\) .


the fibration \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\rightarrow \operatorname{\bf {RPC}}\)  sends an object \((\Gamma /\sigma , D)\)  into \(\sigma \)  and a morphism \((f,\pi ):(\Gamma /\sigma , D)\rightarrow (\Gamma ^{\prime }/\sigma ^{\prime },D^{\prime })\)  into \(f:\sigma \rightarrow \sigma ^{\prime }\) .


Given \(\sigma \in \operatorname{\bf {RPC}}\) , we will denote by \(\mathcal {J}^{trop}_{g,n}(\sigma )\)  the essential fiber of \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\rightarrow \operatorname{\bf {RPC}}\)  over \(\sigma \) , i.e. the groupoid whose objects are \((\Gamma /\sigma , D)\in \mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\)  and whose morphisms are the ones of the form
\(\pi :=(\operatorname{id}_{\sigma }, \pi ): (\Gamma /\sigma , D)\rightarrow (\Gamma ^{\prime }/\sigma ,D^{\prime })\)  in \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\) . Note that any morphism \(\pi :(\Gamma /\sigma , D)\rightarrow (\Gamma ^{\prime }/\sigma ,D^{\prime })\)  in \(\mathcal {J}^{trop}_{g,n}(\sigma )\)  is such that the underlying morphism of graphs \(\pi :(\Gamma ^{\prime })\rightarrow (\Gamma )\)  is an isomorphism.

The universal tropical Jacobian (over \(\operatorname{\bf {RPCC}}\) ), denoted by \(\mathcal {J}^{trop}_{g,n}\) , is the unique stack over \(\operatorname{\bf {RPCC}}\)  (by [1]}) whose restriction to \(\operatorname{\bf {RPC}}\)  coincides with \(\mathcal {J}^{trop}_{g,n}(\operatorname{\bf {RPC}})\) .

